The Solow growth model is built around the neoclassical aggregate produc-

tion function (11.16) and focuses on the proximate causes of growth:

Y = At F( K , L) (11.16)

where Y is real output, K is capital, L is the labour input and At is a measure of

technology (that is, the way that inputs to the production function can be

transformed into output) which is exogenous and taken simply to depend on

time. Sometimes, At is called ˜total factor productivity™. It is important to be

clear about what the assumption of exogenous technology means in the

Solow model. In the neoclassical theory of growth, technology is assumed to

be a public good. Applied to the world economy this means that every

country is assumed to share the same stock of knowledge which is freely

available; that is, all countries have access to the same production function.

In his defence of the neoclassical assumption of treating technology as if it

were a public good, Mankiw (1995) puts his case as follows:

604 Modern macroeconomics

The production function should not be viewed literally as a description of a

speci¬c production process, but as a mapping from quantities of inputs into a

quantity of output. To say that different countries have the same production

function is merely to say that if they had the same inputs, they would produce the

same output. Different countries with different levels of inputs need not rely on

exactly the same processes for producing goods and services. When a country

doubles its capital stock, it does not give each worker twice as many shovels.

Instead, it replaces shovels with bulldozers. For the purposes of modelling eco-

nomic growth, this change should be viewed as a movement along the same

production function, rather than a shift to a completely new production function.

As we shall see later (section 11.15), many economists disagree with this

approach and insist that there are signi¬cant technology gaps between na-

tions (see Fagerberg, 1994; P. Romer, 1995). However, to progress with our

examination of the Solow model we will continue to treat technology as a

public good.

For simplicity, let us begin by ¬rst assuming a situation where there is no

technological progress. Making this assumption of a given state of technol-

ogy will allow us to concentrate on the relationship between output per

worker and capital per worker. We can therefore rewrite (11.16) as:

Y = F( K , L) (11.17)

The aggregate production function given by (11.17) is assumed to be ˜well

behaved™; that is, it satis¬es the following three conditions (see Inada,

1963; D. Romer, 2001; Barro and Sala-i-Martin, 2003; Mankiw, 2003).

First, for all values of K > 0 and L > 0, F(·) exhibits positive but diminish-

ing marginal returns with respect to both capital and labour; that is, ‚F/‚K

> 0, ‚2F/‚K2 < 0, ‚F/‚L > 0, and ‚2F/‚L2 < 0. Second, the production

function exhibits constant returns to scale such that F (»K, »L) = »Y; that

is, raising inputs by » will also increase aggregate output by ». Letting »

=1/L yields Y/L = F (K/L). This assumption allows (11.17) to be written

down in intensive form as (11.18), where y = output per worker (Y/L) and k

= capital per worker (K/L):

y = f ( k ), where f ′( k ) > 0, and f ′′( k ) < 0 for all k (11.18)

Equation (11.18) states that output per worker is a positive function of the

capital“labour ratio and exhibits diminishing returns. The key assumption of

constant returns to scale implies that the economy is suf¬ciently large that

any Smithian gains from further division of labour and specialization have

already been exhausted, so that the size of the economy, in terms of the

labour force, has no in¬‚uence on output per worker. Third, as the capital“

labour ratio approaches in¬nity (k’∞) the marginal product of capital (MPK)

The renaissance of economic growth research 605

y

y = f (k)

k

Figure 11.3 The neoclassical aggregate production function

approaches zero; as the capital“labour ratio approaches zero the marginal

product of capital tends towards in¬nity (MPK’∞).

Figure 11.3 shows an intensive form of the neoclassical aggregate produc-

tion function that satis¬es the above conditions. As the diagram illustrates,

for a given technology, any country that increases its capital“labour ratio

(more equipment per worker) will have a higher output per worker. However,

because of diminishing returns, the impact on output per worker resulting

from capital accumulation per worker (capital deepening) will continuously

decline. Thus for a given increase in k, the impact on y will be much greater

where capital is relatively scarce than in economies where capital is relatively

abundant. That is, the accumulation of capital should have a much more

dramatic impact on labour productivity in developing countries compared to

developed countries.

The slope of the production function measures the marginal product of

capital, where MPK = f(k + 1) “ f(k). In the Solow model the MPK should be

much higher in developing economies compared to developed economies. In

an open economy setting with no restrictions on capital mobility, we should

therefore expect to see, ceteris paribus, capital ¬‚owing from rich to poor

countries, attracted by higher potential returns, thereby accelerating the proc-

ess of capital accumulation.

606 Modern macroeconomics

The consumption function

Since output per worker depends positively on capital per worker, we need to

understand how the capital“labour ratio evolves over time. To examine the

process of capital accumulation we ¬rst need to specify the determination of

saving. In a closed economy aggregate output = aggregate income and com-

prises two components, namely, consumption (C) and investment (I) = Savings

(S). Therefore we can write equation (11.19) for income as:

Y =C+I (11.19)

Y =C+S

or equivalently

Here S = sY is a simple savings function where s is the fraction of income

saved and 1 > s > 0. We can rewrite (11.19) as (11.20):

Y = C + sY (11.20)

Given the assumption of a closed economy, private domestic saving (sY) must

equal domestic investment (I).

The capital accumulation process

A country™s capital stock (Kt) at a point in time consists of plant, machinery

and infrastructure. Each year a proportion of the capital stock wears out. The

parameter δ represents this process of depreciation. Countering this tendency

for the capital stock to decline is a ¬‚ow of investment spending each year (It)

that adds to the capital stock. Therefore, given these two forces, we can write

an equation for the evolution of the capital stock of the following form:

Kt +1 = It + (1 ’ δ ) Kt = sYt + Kt ’ δKt (11.21)

Rewriting (11.21) in per worker terms yields equation (11.22):

Kt +1 / L = sYt / L + Kt / L ’ δKt / L (11.22)

Deducting Kt /L from both sides of (11.22) gives us (11.23):

Kt +1 / L ’ Kt / L = sYt / L ’ δKt / L (11.23)

In the neoclassical theory of growth the accumulation of capital evolves

according to (11.24), which is the fundamental differential equation of the

Solow model:

k = sf ( k ) ’ δk

™ (11.24)

The renaissance of economic growth research 607

™

where k = Kt+1 /L “ Kt /L is the change of the capital input per worker, and

sf(k) = sy = sYt /L is saving (investment) per worker. The δk= δKt /L term

represents the ˜investment requirements™ per worker in order to keep the

capital“labour ratio constant. The steady-state condition in the Solow model

is given in equation (11.25):